Most branches of mathematics have big, sexy famous open problems. Number theory has the Riemann hypothesis and the Langlands program, among many others. Geometry had the Poincaré conjecture for a long time, and currently has the classification of 4-manifolds. PDE theory has the Navier-Stokes equation to deal with.

So what are the big problems in probability theory and stochastic analysis? I'm a grad student working in the field, but I can't name any major unsolved conjectures or open problems which are driving research. I've heard that stochastic Löwner evolutions are a big field of study these days, but I don't know what the conjectures or problems relating to them are.

Understanding SAW is certainly one of the biggest outstanding problems in probability theory. Nonetheless, it's premature to select it as The Answer within hours of posting your question.
–
Tom LaGattaAug 30 '10 at 18:50

6

Since it is not CW, this means there is one, unique, answer. So this must be it!
–
Gerald EdgarAug 30 '10 at 21:39

To my mind the sexiest of open problems in probability is to show that there is "no percolation at the critical point" (mentioned in particular in section 4.1 of Gordon Slade's contribution to the Princeton Companion to Mathematics). A capsule summary: write $\mathbb{Z}_{d,p}$ for the random subgraph of the nearest-neighbour $d$-dimensional integer lattice, obtained by independently keeping each edge with probability $p$. Then it is known that there exists a critical probability $p_c(d)$ (the percolation threshold}) such that for $p < p_c$, with probability one
$\mathbb{Z}_{d,p}$ contains no infinite component, and for $p > p_c$, with probability one there exists an unique infinite component.

The conjecture is that with probability one, $\mathbb{Z}_{d,p_c(d)}$ contains no infinite component. The conjecture is known to be true when $d =2$ or $d \geq 19$.

Incidentally, one of the most effective ways we have of understanding percolation -- a technique known as the lace expansion, largely developed by Takeshi Hara and Gordon Slade -- is also one of the key tools for studying self-avoiding walks and a host of other random lattice models.

That article of Slade's is in fact full of intriguing conjectures in the area of critical phenomena, but the conjecture I just mentioned is probably the most famous of the lot.

I agree that this conjecture (referred to as the dying percolation conjecture) is a great open problem. It is especially challenging in dimensions 3,4, and 5 where the Hara Slade results do not hold.
–
Gil KalaiAug 30 '10 at 21:56

In the $n$-dimensional grid, start with a vertex colored black and all others colored white. Choose uniformly a bicolor edge (one black end, one white end) and color in black its white end. Continue this process forever.

The black part grows, and it is known that if we rescale it so that it has constant diameter, it converges to a convex shape. What we do not know is what the shape is.

What Benoît has described is the Richardson growth model, which has limiting shape equal to that of first-passage percolation with i.i.d. exponential passage times. What is most fascinating to me is that the limiting shape is not known for any distribution of i.i.d. passage times. There are related models for which the limiting shape is known (e.g. last-passage percolation, Euclidean FPP, FPP with stationary and ergodic passage times), but the i.i.d. case has resisted all attack.
–
Tom LaGattaAug 30 '10 at 19:01

1

Another variation (the general Richardson model) is to choose some $p$, and for each boundary edge to color its white end black with probability $p$. The limit shape is not known except obviously for $p=1$, and when $p\to 0$ the limit shape converges to the limit shape of first passage percolation. An interesting fact is known, though: if $p$ is close enough to $1$, then the limit shape is not strictly convex.
–
Benoît KloecknerAug 31 '10 at 8:15

David Aldous has a list of open problems on his website, though they look like personal favorites rather than "big" questions. You might look at the problems Aldous labels as "Type 2:We have a precise mathematical problem, but we do not see any plausible outline for a potential proof."

In an earlier version (stat.berkeley.edu/~aldous/Research/problems.ps) of that list of open problems, Aldous states that 'they are not intended to be "representative" or "the most important" ... of all open problems in probability. The majority are (I think) my own invention and have not been discussed extensively elsewhere'. That having been said, I really enjoy Aldous' list and find many of his open problems dangerously fun to think about.
–
Louigi Addario-BerryAug 30 '10 at 19:44

Michel Talagrand has a number of open problems (with bounty) listed on his website. I haven't looked at them all, but knowing him, I guarantee you that they are very hard and quite important. These are motivated by his research directions, but unlike some fields, there's not one research direction and one set of open problems that dominate probability theory right now.

One major problem is extending the wonderful understanding of planar stochastic models to higher dimensions. So understanding 3,4-dimensional percolation, Ising Model, self avoiding walks, loop erased random walks and their scaling limits is a rather important problem.

The lack of a so-called big problem in probability theory seems to suggest the richness of the subject itself. One of the most fascinating subfields is the determination of convergence rate of finite state space Markov chains. Many convergence problem even on finite groups have exhausted current analytic techniques. For instance, intuitions from coupon collector's problem suggests that the random adjacent transposition walk exhibits cutoff in total variation convergence to the uniform measure on the symmetric group, and the upper and lower bounds gap is only a factor of 2. There are many tools one could employ to study such problems, such as representation theory and discretized version of inequalities from PDE theory, which makes the solutions very creative.

Number theory has several big problems but is also a very rich subject (not all work in number theory is directed towards the Riemann hypothesis or the Birch and Swinnerton-Dyer conjecture), so the lack of a big problem in probability does not really point to the subject's richness.
–
KConradFeb 2 '11 at 17:30

1

I suppose if you couple it with the fact so many people work in it, then richness does become a corollary.
–
John JiangFeb 2 '11 at 19:55

I wasn't intending to suggest probability is not a rich subject, but I don't buy the argument that the lack of a very prominent unsolved question or program in an area is in some way a sign that the area is rich. Seems kind of after-the-fact justification to me.
–
KConradApr 15 '14 at 21:17

The normal distribution and the many places it occurs in mathematics and its application is a primary example of a universal phenomenon. Proving and understanding other universal phenomena in probability is of great importance. One example I like is to understand the distributions that came from random matrix theory and occur in various other places. One such distribution is the distribution of the largest eigenvalue of a random matrix discovered by Tracy and Widom.

Maybe the no1 problem of probability is to make rigorous what one finds in just about any textbook in statistical mechanics. In other words it is to put the predictions of Wilson's renormalization group theory on a rigorous footing. Many of the topics mentioned in this post are particular conjectures in this broader program.

Peligrad showed the result holds with the assumption $\liminf_{n\to +\infty}n^{-1}\operatorname{Var}(S_n)>0$. It also holds when $\mathbb E\lvert X_0\rvert^{2+\delta}$ is finite for some positive $\delta$ (Ibragimov, I think).